How Many Marbles?
... to happen. Say: “We name a probability with a number from 0 to 1. If an event is certain to happen, then the probability is 1. If an event is certain not to happen, the probability is 0. If it may or may not happen, then the probability is some fraction between 0 and 1.” 2. Make a number line on the ...
... to happen. Say: “We name a probability with a number from 0 to 1. If an event is certain to happen, then the probability is 1. If an event is certain not to happen, the probability is 0. If it may or may not happen, then the probability is some fraction between 0 and 1.” 2. Make a number line on the ...
Hitting 10
... One more question––can we work out P100 from this? Of course we could start with P1 and iterate the recursion 99 times. That would be easy enough with a spreadsheet, but where would it get us? What we’d really like is a formula for PN in terms of N. And that’s really a question of solving the recur ...
... One more question––can we work out P100 from this? Of course we could start with P1 and iterate the recursion 99 times. That would be easy enough with a spreadsheet, but where would it get us? What we’d really like is a formula for PN in terms of N. And that’s really a question of solving the recur ...
Normal numbers without measure theory - Research Online
... quite accesible to undergraduate students, except at one point, where it is necessary to invoke the Beppo Levi Theorem to interchange the order of summation and integration in a series of non-negative functions. The Beppo Levi Theorem is similarly invoked by Goodman [1], in his extension of Kac’s ap ...
... quite accesible to undergraduate students, except at one point, where it is necessary to invoke the Beppo Levi Theorem to interchange the order of summation and integration in a series of non-negative functions. The Beppo Levi Theorem is similarly invoked by Goodman [1], in his extension of Kac’s ap ...
some applications of probability generating function based methods
... Step 1. Choose a value for the unknown parameter θ. Let j = 1. Choose r the number of repetitions of the simulation. Step 2. Simulate a sample of X(θ). Step 3. Determine by the method described above θ̂ 1,j an estimated value of θ and, by another standard and known method, θ̂ 2,j another estimated v ...
... Step 1. Choose a value for the unknown parameter θ. Let j = 1. Choose r the number of repetitions of the simulation. Step 2. Simulate a sample of X(θ). Step 3. Determine by the method described above θ̂ 1,j an estimated value of θ and, by another standard and known method, θ̂ 2,j another estimated v ...
Random Number Generation
... observations from probability distributions. 1. Random number generation. 2. Random variate generation. ...
... observations from probability distributions. 1. Random number generation. 2. Random variate generation. ...
Probabilistic Theories of Type
... Formally: Let LTr be the first-order language of arithmetic extended by Tr . (For convenience, we will use the standard model of arithmetic as our “ground model”; we also fix a recursive coding scheme for LTr .) Question: Is there a function P : LTr → [0, 1], such that: P satisfies the axioms of pr ...
... Formally: Let LTr be the first-order language of arithmetic extended by Tr . (For convenience, we will use the standard model of arithmetic as our “ground model”; we also fix a recursive coding scheme for LTr .) Question: Is there a function P : LTr → [0, 1], such that: P satisfies the axioms of pr ...
Constructing k-wise Independent Variables
... The above bound is non-trivial only for q = o(log log n). Can we do better? Yes, we were too generous with the number of pairwise independent functions over {0, 1}q . For now, let’s consider the equivalent set of functions, {f : {0, 1}q → {−1, 1}}. We can associate with each q such function f a vect ...
... The above bound is non-trivial only for q = o(log log n). Can we do better? Yes, we were too generous with the number of pairwise independent functions over {0, 1}q . For now, let’s consider the equivalent set of functions, {f : {0, 1}q → {−1, 1}}. We can associate with each q such function f a vect ...
Lecture notes on the pigeonhole principle and
... of this equation. If we do this, then we get two disjoint subsets which both have the same sum. There is just one last thing to check, which is that we don’t get the equation 0 = 0 after eliminating the common variables. This cannot happen since the original two subsets were different. One more comm ...
... of this equation. If we do this, then we get two disjoint subsets which both have the same sum. There is just one last thing to check, which is that we don’t get the equation 0 = 0 after eliminating the common variables. This cannot happen since the original two subsets were different. One more comm ...
Probability and Random Processes Measure
... • For T from (Ω, A) to (Λ, S), • the class T −1 (S) is a σ-algebra ⊂ A • the class {L ⊂ Λ : T −1 (L) ∈ A} is a σ-algebra ⊂ S • if S = σ(C) for some C, then T is measurable (from A to S) ...
... • For T from (Ω, A) to (Λ, S), • the class T −1 (S) is a σ-algebra ⊂ A • the class {L ⊂ Λ : T −1 (L) ∈ A} is a σ-algebra ⊂ S • if S = σ(C) for some C, then T is measurable (from A to S) ...
NZMATH Users Manual
... After the creation of Python 2.3, it was theoretically possible to use random.randrange, since it started to accept long integer as its argument. Use of it was, however, not considered, since there had been the bigrandom module. It was lucky for us. In fall of 2006, we found a bug in random.randrang ...
... After the creation of Python 2.3, it was theoretically possible to use random.randrange, since it started to accept long integer as its argument. Use of it was, however, not considered, since there had been the bigrandom module. It was lucky for us. In fall of 2006, we found a bug in random.randrang ...
ON THE NUMBER OF VERTICES OF RANDOM CONVEX POLYHEDRA 1 Introduction
... with the vectors ai for which k + 1 ≤ i ≤ r and i ∈ J, maintaining the condition that the first r vectors are linearly independent. If such an interchange is not possible because the first r vectors would be linearly dependent then every aj , r + 1 ≤ j ≤ n, j ∈ / J would be an element of the subspace ...
... with the vectors ai for which k + 1 ≤ i ≤ r and i ∈ J, maintaining the condition that the first r vectors are linearly independent. If such an interchange is not possible because the first r vectors would be linearly dependent then every aj , r + 1 ≤ j ≤ n, j ∈ / J would be an element of the subspace ...
Derivation of Binomial Probability Formula
... “What is the probability of having the two yellow peas occur in the first or the second or the third or the fourth or the fifth or the sixth or the seventh or the eighth or the ninth or the tenth arrangement listed above?” Again, we may have some grammatical issues with our translation, but the math ...
... “What is the probability of having the two yellow peas occur in the first or the second or the third or the fourth or the fifth or the sixth or the seventh or the eighth or the ninth or the tenth arrangement listed above?” Again, we may have some grammatical issues with our translation, but the math ...
Problem of the Day 1. You have 8 nice shirts, 5 pairs of nice pants
... How many different 4 number phone codes are possible? How many 4 number codes are possible if no number can be repeated? How many different 4 number codes are possible if no adjacent numbers can be the same? The U.S. Senate consists of 100 senators. How many different ways can we form the Se ...
... How many different 4 number phone codes are possible? How many 4 number codes are possible if no number can be repeated? How many different 4 number codes are possible if no adjacent numbers can be the same? The U.S. Senate consists of 100 senators. How many different ways can we form the Se ...
Infinite monkey theorem
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.In this context, ""almost surely"" is a mathematical term with a precise meaning, and the ""monkey"" is not an actual monkey, but a metaphor for an abstract device that produces an endless random sequence of letters and symbols. One of the earliest instances of the use of the ""monkey metaphor"" is that of French mathematician Émile Borel in 1913, but the first instance may be even earlier. The relevance of the theorem is questionable—the probability of a universe full of monkeys typing a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low (but technically not zero). It should also be noted that real monkeys don't produce uniformly random output, which means that an actual monkey hitting keys for an infinite amount of time has no statistical certainty of ever producing any given text.Variants of the theorem include multiple and even infinitely many typists, and the target text varies between an entire library and a single sentence. The history of these statements can be traced back to Aristotle's On Generation and Corruption and Cicero's De natura deorum (On the Nature of the Gods), through Blaise Pascal and Jonathan Swift, and finally to modern statements with their iconic simians and typewriters. In the early 20th century, Émile Borel and Arthur Eddington used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics.